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FIG 10.1. Reaction rate is profoundly influenced by temperature.
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FIG 10.2. Comparison of conversion vs. temperature characteristics for (a) plug-flow and (b) back-mixed reactors.
And for the back-mixed reactor!
dy E dT= pY(l
- Y)
(10.12)
These expressions will be useful in determining temperature stabilit y later in the chapter. The maximum slope of a conversion vs. temperature curve always occurs at kV/F = 1, which corresponds to 63 percent conversion in t he plug-flow reactor and 50 percent in the back-mixed. For the V/F = 1 curves of Fig. 10.2, the maximum slopes are 1/64.7 F and 1/95 F for the two reactors, at 230 F. The curves of Fig. 10.2 and Eqs. (10.11) and (10.12) describe st eadystate conditions only, however. In a departure from steady state because of a heat transfer upset, temperature will change the reaction-rate coeflicient in advance of a change in reactant concentration. Thus the reaction rate will increase with temperature above the new steady-st ate level unt il reactant concentration is accordingly reduced. A partial derivative of conversion with respect to temperature at constant concentration describes t he instantaneous conversion that exceeds the steady-state conversion relative to the amount unconverted:2 (10.13) This means that the maximum dynamic slope of the plug-flow reactor is
dT z
aY -
= g2 In (1 - y)
(10.14)
1 Applications
And for the back-mixed reactor, (10.15)
The Stability of Exothermic Reactors
An exothermic reaction is one in which heat is evolved. The evolution of heat increases temperature, which increases the rate of reaction. This series of events forms a positive feedback loop which can result in a runaway if other conditions permit. The conditions are: 1. Heat cannot be removed to the surroundings as fast as it is evolved. 2. Conversion is sufficiently below 100 percent that heat evolution is not thereby limited. The rate of heat evolution Q1- is simply the rate of reaction times the heat of reaction H,:
Qr = H,Fxoy
(10.16)
Because Q7 varies directly with y, the curves of Fig. 10.2 can also be plotted as heat evolution against temperature. Figure 10.3 shows the heat evolution of t he back-mixed reactor at B/F = 1; the temperature is to be controlled at 230 F to maintain 50 percent conversion. For the moment, neglect the sensible heat of the reactants, such that all the evolved heat is to be transferred to a cooling system. The rate of heat transfer Q7, will approach
&TX = UA(T - T,)
where 7 , is the coolant temperature. This describes a straight line of slope rA and intercept l ,. Lines representing two possible cooling systems designed for the same heat flow are also shown in Fig. 10.3. The normal condition for the reactor is described by point 0 in Fig. 10.3. No matter which heat removal line is followed, QT = Qr at that point, so that a state of thermal equilibrium can exist. But should the temperature rise, the rate of heat evolution will increase more than the rate
0 160
FIG 10.3. The slope of the heatremoval line determines whether the reactor will be stable. 200
220 240 Temperature. F 260 280
Controlling Chemical Reactions
of heat transferred by the system designated as unstable.3 This will cause the t,emperature to rise farther until the QV curve crosses the line again, at point IV. Point s I, and N arc stable intersections, while point 0 is an unstable intersection. The other heat transfer line demonstrates a capability for removing more heat than is evolved upon a temperature increase, thereby restoring equilibrium. This indicates that an cxothermic reactor can be made inherently stable by providing sufficient heat transfer area. To state it another way, sufficient heat transfer area will provide negative feedback in excess of the positive feedback of the reaction. To more explicitly define the relationships involved, an unsteady-st ate heat balance must be written: H,Fx,,y - UA (T - T,) - FpC( T - TF) = VpC g (10.17)
The first two terms of Eq. (10.17) have already been described. The third term represents the sensible heat absorbed by t he reaction stream of density p and specific heat C as it rises from inlet temperature TF. The term on the right of the equation represents the thermal capacity of the system. Since heat evolution is a nonlinear function of temperature, it is necessary to linearize Eq. (10.17) in order to find the thermal time constant of the reactor. Let Eq. (10.18) describe variations about a designated reference temperature Z ,: z ( T - kl-) - UA(T - T,) - FpC(T - T,) = vpc$ (10.18)
Arranging in the classical form of a first-order equation allows identification of the time constant:
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